Trigonometry & Inverse Trigonometry
Range and Domain
Grade 11
Question:
<p>The range of \(y = \frac{\sin^4 x - \sin^2 x}{\sin^4 x + \sin^2 x}\) satisfies:</p>
<p>(a) \(y \in \left(-\infty, \frac{1}{3}\right]\)</p>
<p>(b) \(y \in \left[\frac{1}{3}, 1\right)\)</p>
<p>(c) \(y \in (1, 3)\)</p>
<p>(d) \(y \in [3, \infty)\)</p>
Step-by-Step Solution
Key Concept: Substitute \(t = \sin^2 x\) and analyze the resulting rational function for \(t \in [0,1]\).
<p>Let \(t = \sin^2 x\) where \(0 \leq t \leq 1\). Then \(y = \frac{t^2 - t}{t^2 + t} = \frac{t(t-1)}{t(t+1)} = \frac{t-1}{t+1}\). Since \(0 \leq t \leq 1\), we get \(y \in [-1, 0]\). However, when \(\sin x = 0\), the expression is undefined. Analyzing the extremes: as \(t \to 0^+\), \(y \to -1\); as \(t \to 1\), \(y \to 0\). The range includes both negative values and approaches \(0\).</p>
Correct Answer: a, d